I started thinking about how the relationship between your dominant and your non dominant eye might influence your aim in FPS games so I looked up a study and found some interesting points. Here's the most relevant part of the conclusion.
"Our main finding is that ocular dominance reverses as a function of ocular dominance, with some modulation determined by the hand used. The functional significance of these results seems to be straightforward. Although the binocular field of view is ≈100° when looking straight ahead, as the eyes rotate peripherally, the monocular field of the inward turning eye is increasingly occluded (largely by the nose) by up to 50%. Therefore, it makes sense for the eye–hand coordination system to choose the eye with best overall field of view. At the same time, in most subjects this gating function is shifted slightly to the left or right (Fig. 2a), most likely to avoid dominance ‘flickering’ at the commonplace central range (i.e. the range tested in previous studies). This strategy allows for preferential gating of visual input from the eye with the best field of view, while avoiding ambiguity at the most common central gaze position."
Basically, the main finding was that people's dominant eye shifts based on what degree they're looking at. They also stated that the switch in dominance is pretty abrupt so Its pretty easy to see how this could have an impact on somebody's ability to aim in TF2 when you consider how players constantly shift their angle of view while tracking and therefor shift their eye dominance. If you're feeling skeptical about the relevancy, I would like to point out that the distance of the target stimuli in the tests were about 20 inches away from the person, which is pretty similar to how close we sit to our monitors and that the threshold for dominance to switch is about 10-20 degrees horizontally, which we certainly meet while tracking fast moving objects on our monitors.
If our dominant eye is shifting, so does our perceived position of an object. This leads me to believe that if we position our eyes in a way that minimizes the likelihood of switching eye dominance, we will have better aim. Since most people are right eye dominant, if you angle your head slightly to the left you'll have to shift your eyes much further to the left to in order to reach the the threshold needed to switch dominance. You could also shift the position of your monitor in the direction of your dominant eye.
EDIT: link disappeared when I edited the post.
http://www.sciencedirect.com/science/article/pii/S0042698901000797